Applications of the Fuglede-Kadison determinant: Szegö’s theorem and outers for noncommutative 𝐻^{𝑝}

Blecher, David; Labuschagne, Louis

doi:10.1090/S0002-9947-08-04506-6

Applications of the Fuglede-Kadison determinant: Szegö’s theorem and outers for noncommutative $H^p$
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Trans. Amer. Math. Soc. 360 (2008), 6131-6147 Request permission

Abstract:

We first use properties of the Fuglede-Kadison determinant on $L^p(M)$, for a finite von Neumann algebra $M$, to give several useful variants of the noncommutative Szegö theorem for $L^p(M)$, including the one usually attributed to Kolmogorov and Krein. As an application, we solve the longstanding open problem concerning the noncommutative generalization, to Arveson’s noncommutative $H^p$ spaces, of the famous ‘outer factorization’ of functions $f$ with $\log |f|$ integrable. Using the Fuglede-Kadison determinant, we also generalize many other classical results concerning outer functions.

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